On a rimmed baking sheet, toss tomatoes and shallot; season with 1 teaspoon salt and 1 teaspoon pepper. Drizzle oil and vinegar over. Let sit until tomatoes have released their liquid and shallots are softened, about 30 minutes.
In a small skillet over low heat, toast panko, stirring occasionally, until golden brown, about 2 minutes. Transfer to a small bowl.
Layer tomatoes on a large platter. Spoon shallot and juices over top. Sprinkle with panko. Arrange burrata in center of tomatoes and crack open with your hands. Sprinkle with basil, chives, salt, and pepper. Serve with bread alongside.
